scholarly journals On the Discontinuity of the Costates for Optimal Control Problems With Coulomb Friction

2000 ◽  
Author(s):  
Brian J. Driessen ◽  
Nader Sadegh
Keyword(s):  
Optimal Control ◽  
Coulomb Friction ◽  
Optimal Control Problems ◽  
Minimum Time ◽  
Control Problems ◽  
Time Points ◽  
Discontinuous Functions ◽  
Time Problem ◽  
Switching Functions ◽  
Minimum Time Problem

Abstract This work points out that the costates are actually discontinuous functions of time for optimal control problems with Coulomb friction. In particular these discontinuities occur at the time points where the velocity of the system changes sign. To our knowledge, this has not been noted before. This phenomenon is demonstrated on a minimum-time problem with Coulomb friction and the consistency of discontinuous costates and switching functions with respect to the input switches is shown.

Improved Gradient-Type Algorithms for Zero Terminal Gradient Optimal Control Problems

1987 ◽  
Vol 109 (4) ◽  
pp. 355-362
Author(s):  
Chung-Feng Kuo ◽  
Chen-Yuan Kuo
Keyword(s):  
Optimal Control ◽  
Rate Of Convergence ◽  
Optimal Control Problems ◽  
Control Variable ◽  
Transformation Method ◽  
Control Problems ◽  
Final Time ◽  
Time Problem ◽  
Gradient Type ◽  
Improved Algorithm

Difficulties often arise when we apply the gradient type algorithms employing penalty functions to optimal control problems with variable final time. There is another class of optimal control problems for which the necessary conditions for optimality require a zero gradient at the final time. This causes the gradient-type algorithms, in their standard forms, to become incapable of changing the terminal value of the control variable at each iteration and the rate of convergence is adversely affected. In this paper, we first apply a new transformation method developed by Polak [19] which transforms the variable final time problem into a fixed final time problem. Second, an improved gradient-type algorithm is developed to overcome the zero terminal gradient problem. It is shown that, by applying this transformation and improved algorithm to four examples, not only the variable final time and zero terminal gradient problems are solved and the control vector updated in the correct direction but the rate of convergence of the improved algorithm is faster than that of the traditional gradient-type algorithms.

scholarly journals Numerical solution of an optimal control problem with variable time points in the objective function

2002 ◽  
Vol 43 (4) ◽  
pp. 463-478 ◽  
Author(s):  
K. L. Teo ◽  
Y. Liu ◽  
W. R. Lee ◽  
L. S. Jennings ◽  
S. Wang
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Control Problems ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Control Problems ◽  
Time Points ◽  
Variable Time ◽  
Control Functions

AbstractIn this paper, we consider the numerical solution of a class of optimal control problems involving variable time points in their cost functions. The control enhancing transform is first used to convert the optimal control problem with variable time points into an equivalent optimal control problem with fixed multiple characteristic time (MCT). Using the control parametrization technique, the time horizon is partitioned into several subintervals. Let the partition points also be taken as decision variables. The control functions are approximated by piecewise constant or piecewise linear functions in accordance with these variable partition points. We thus obtain a finite dimensional optimization problem. The control parametrization enhancing control transform (CPET) is again used to convert approximate optimal control problems with variable partition points into equivalent standard optimal control problems with MCT, where the control functions are piecewise constant or piecewise linear functions with pre-fixed partition points. The transformed problems are essentially optimal parameter selection problems with MCT. The gradient formulae for the objective function as well as the constraint functions with respect to relevant decision variables are obtained. Numerical examples are solved using the proposed method.

Solution of optimal control problems on a high-speed hybrid computer

SIMULATION ◽  
1966 ◽  
Vol 7 (5) ◽  
pp. 238-245 ◽  
Author(s):  
Richard L. Maybach
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
High Speed ◽  
Optimal Control Problems ◽  
Initial Conditions ◽  
The State ◽  
Minimum Time ◽  
Adjoint Equations ◽  
Control Problems ◽  
Hybrid Computer

This paper presents a method for finding the solutions to minimum-time optimal control problems. The procedure is to implement Pontryagin's Maximum Principle on an iterative hybrid computer. The state and adjoint equations as well as the control law are simulated using conven tional analog components. The troublesome two-point boundary-value problem, which is always associated with Maximum Principle, is solved by iteration, using a digital parameter optimizer. Thus, a manual trial-and-error search for the proper initial values of the adjoint variables is un necessary. We show that, for a large class of systems, it is not necessary to generate the Hamiltonian, because the neces sary condition that it normally must satisfy is redundant. This allows many problems to be greatly simplified. We also present an optimizing routine that solves the boun dary-value problem. This permits the proposed method to be used on any hybrid computer that incorporates a general-purpose digital computer. The solutions to two problems show that the proposed method is feasible. Average convergence times range from less than one second to about 70 seconds. These vary with the initial conditions on the state variables. The examples were solved using ASTRAC II, a small (40 amplifier), high speed (up to 1000 solutions per second), iterative hybrid computer with only modest component accuracy (0.25 per cent). Although the discussion and examples are limited to a minimum-time performance index, the method is easily extended to cover other criteria.

MILP formulation for solving minimum time optimal control problems

1990 ◽  
Vol 51 (4) ◽  
pp. 943-947 ◽  
Author(s):  
F. D. CARVALLO† ◽  
A. W. WESTERBERG† ◽  
M. MORARI†
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Minimum Time ◽  
Time Optimal Control ◽  
Control Problems ◽  
Time Optimal
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